[leftmargin=15pt]
Mann–Whitney U [1]}, [2]}.
This test statistic, \(U = \#{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}\)  estimates how much \(P(Y<X)\)  deviates from \(1/2\)  for randomly selected \(X\)  and \(Y\) . The equivalent combination \(u = ({\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}- {\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}})/2\)  lies in the component \(W_{\kappa 1}\)  where \(\kappa =(1,1)\) . The null distribution of \(\tilde{\#}u\)  is asymptotically normal with variance \({\tfrac{1}{12}}(\tfrac{1}{m} + \tfrac{1}{n})\) , reproduced by Theorems REF -REF .

Cramér–von Mises criterion [3]}.
The above \(U\)  might fail to detect \(X \lnot \sim Y\)  when the probability of \(X<Y\)  happens to be exactly \(1/2\) . However, given four independent replications \(X\) , \(X^{\prime }\) , \(Y\) , and \(Y^{\prime }\) , the probability that \(\max (X,X^{\prime })<\min (Y,Y^{\prime })\)  or \(\max (Y,Y^{\prime })<\min (X,X^{\prime })\)  is \(1/3\)  if and only if \(X \sim Y\) . Otherwise, it is greater than \(1/3\)  by an \(L^2\)  difference between the distribution functions \(F_X\)  and \(F_Y\) . This difference can be estimated by \(2\,\tilde{\#}t\)  for the following centralized combination in \(W_{(2,2)}\) :
\( t \;=\; \tfrac{1}{3}({\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily y}}+ {\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily x}}) - \tfrac{1}{6}({\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}+ {\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}+ {\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}+ {\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}) \) 
Theorems REF -REF  give \(t \in W_{(2,2)201}\)  and \(\operatorname{V}[\tilde{\#}t] \;\sim \; \tfrac{1}{45}(\tfrac{1}{m} + \tfrac{1}{n})^2\)  in agreement with [4]}.

Watson's \(U^2\)  [5]}.
Now suppose that \(\lbrace X_i\rbrace \)  and \(\lbrace Y_i\rbrace \)  are samples on the circle \(S^1\) . In this case, the previous test for \(X \sim Y\)  depends on an arbitrary choice of a starting point. Another notion of difference by Watson can be estimated by \(\tilde{\#}s\) , for the following rotation invariant combination.
\( s \;=\; \tfrac{1}{12}({\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily y}}+ {\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily x}}+ {\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}+ {\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}) - \tfrac{1}{6}({\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}+ {\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}{\text{\bfseries \ttfamily y}}{\text{\bfseries \ttfamily x}}) \) 
This is not a principal direction of the covariance, but \(s = v + \tfrac{1}{4} t\)  for \(v \in W_{(2,2)200}\) , and in fact \(W_{(2,2)2} = \operatorname{span}\lbrace s,t\rbrace \) . By Theorem REF , \(\operatorname{V}[\tilde{\#}v] \sim \tfrac{1}{720}(\tfrac{1}{m} + \tfrac{1}{n})^2\) , so \(\operatorname{V}[\tilde{\#}s] \sim \tfrac{1}{360}(\tfrac{1}{m} + \tfrac{1}{n})^2\) .

